Alternatives to TSLS

There are several alternatives to the standard IV/TSLS estimator. Among them is the limited information maximum likelihood (LIML) estimator, which was first derived by Anderson and Rubin(1949). There is renewed interest in LIML because evidence indicates that it performs better than TSLS when instruments are weak. Several modifications of LIML have been suggested by Fuller(1977) and others. These estimators are unified in a common framework, along with TSLS, using the idea of a k-class of estimators. LIML suffers less from test size aberrations than the TSLS estimator, and the Fuller modification suffers less from bias. Each of these alternatives will be considered below.

In a system of M simultaneous equations let the endogenous variables be y1,y2,… ,yM. Let there be K exogenous variables x1,x2,… ,xK. The first structural equation within this system is

which is consistently estimated by least squares. The predictions from the reduced form are

E (У2) = П12Х1 + П22Х2 +—- + ПК2ХК (11.8)

and the residuals are v2 = y2 — E (y2).

The two-stage least squares estimator is an IV estimator using E (y2) as an instrument. A k-class estimator is an IV estimator using instrumental variable y2 — kv2. The LIML estimator uses k = l where I is the minimum ratio of the sum of squared residuals from two regressions. The explanation is given on pages 468-469 of POE4. A modification suggested by Fuller (1977) that uses the k-class value

(11.9)

where K is the total number of instrumental variables (included and excluded exogenous variables) and N is the sample size. The value of a is a constant-usually 1 or 4. When a model is just identified, the LIML and TSLS estimates will be identical. It is only in overidentified models that the two will diverge. There is some evidence that LIML is indeed superior to TSLS when instruments are weak and models substantially overidentified.

A script can be used to estimate the model via LIML. The following one is used to replicate the results in Table 11B.3 of POE4.

1 open "@gretldirdatapoemroz. gdt"

2 square exper

3 series nwifeinc = (faminc-wage*hours)/1000

4 smpl hours>0 —restrict

5 list x = mtr educ kidsl6 nwifeinc const

6 list z1 = educ kidsl6 nwifeinc const exper

7 list z2 = educ kidsl6 nwifeinc const exper sq_exper largecity

8 list z3 = kidsl6 nwifeinc const mothereduc fathereduc

9 list z4 = kidsl6 nwifeinc const mothereduc fathereduc exper

10

10 tsls hours x; z1 –liml

11 tsls hours x; z2 –liml

12 tsls hours x; z3 –liml

13 tsls hours x; z4 –liml

LIML estimation uses the tsls command with the –liml option. The results from LIML estima­tion of the hours equation, (11.10) the fourth model in line 14, are given below. The variables mtr and educ are endogenous, and the external instruments are mothereduc, fathereduc, and exper; two endogenous variables with three external instruments suggests that the model is overidentified in this specification.